72.3. THE SITUATION 2443

Assume Y (s) satisfies

Y ∈ K′ = Lp′ ([0,T ]×Ω;V ′)

where 1/p′+ 1/p = 1 and Y is V ′ progressively measurable. The situation in which theequation holds is as follows. For a.e. ω, the equation holds for all t ∈ [0,T ] in V ′. Thusit follows that X (t) is automatically progressively measurable into V ′ from Proposition72.1.2. Also W (t) is a Wiener process on U1 in the above diagram. Thus X is continuousinto V ′ off a set of measure zero, and it is also V ′ predictable.

The goal is to prove the following Ito formula.

|X (t)|2 = |X0|2 +∫ t

0

(2⟨Y (s) , X̄ (s)⟩+ ||Z (s)||2

L2(Q1/2U,H)

)ds

+2∫ t

0R((

Z (s)◦ J−1)∗ X̄ (s))◦ JdW (s) (72.3.3)

where R is the Riesz map which takes U1 to U ′1. The main thing is that the last term abovebe a local martingale.

In all that follows, the mesh points t j will be points where X̄ (t j) = X (t j) a.e. ω .

Lemma 72.3.2 Let X be as in Situation 72.3.1 and let X lk be as in Lemma 72.2.1 corre-

sponding to X̄ above. Say

X lk (t) =

mk

∑j=0

X̄ (t j)X[t j ,t j+1)(t) , X l

k (0)≡ 0.

Then each term in the above sum for which t j > 0 is predictable into H. As mentionedearlier, we can take X (0) ≡ 0 in the definition of the “left step function”. Since, at themesh points, X̄ = X a.e., it makes no difference off a set of measure zero whether we useX̄ (t j) or X (t j) at the left end point.

Proof: This is a step function and a typical term is of the form X (a)X[a,b) (t) . I willtry and show this is predictable. Let an be an increasing sequence converging to a and letbn be an increasing sequence converging to b. Then for a.e. ω,

X (an)X(an,bn] (t)→ X (a)X[a,b) (t)

in V ′ due to the fact that t→ X (t) is continuous into V ′ for a.e. ω . Therefore, letting v ∈Vbe given, it follows that for a.e. ω⟨

X (an)X(an,bn] (t) ,v⟩→⟨X (a)X[a,b) (t) ,v

⟩,

and since the filtration is a normal filtration in which all sets of measure zero from FT arein F0, this shows

(t,ω)→⟨X (a)X[a,b) (t) ,v

⟩